Page 268 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
=B13 (a \ [(a \ (b \ c)]) \ (c \ [a \ (b \ c)]) =B8 (a \ c) \ (a \ (b \ c))
=B7 (a \ [a \ (b \ c)]) \ c =B8 (a \ c) \ ([(a \ (b \ c)] \ c)
=B7 (a \ c) \ [(a \ c) \ (b \ c)] =B5 (a \ c) ∧ (b \ c) =B18 (a \ c) \ (a \ b). £
This leads us to:
Theorem 7.1.6. Given an iBCS-algebra (A; \, 0), upon setting x∧y = x \ (x \ y) the
derived algebra (A; ∧, 0) is a left normal band with zero.
Proof. B1 follows from a ∧ a = a \ (a \ a) = a \ 0 = a. B4 follows from
a ∧ 0 = a \ (a \ 0) = a \ a = 0 and 0 ∧ a = 0 \ (0 \ a) = 0
provided we know that 0 \ a = 0 holds in general. But the latter follows from 0 \ a = (a \ a) \ a =
(a \ a) \ (a \ a) = 0 \ 0 = 0. To verify B2 and B3 we again follow Spinks [2002]. To begin,
observe that
(a ∧ b) ∧ c =B5 (a ∧ b) \ [(a ∧ b) \ c] =B5, 16 [a \ (a \ b)] \ [(a \ c) \ (a \ b)]
=B8 [a \ (a \ c)] \ (a \ b) =B7 [a \ (a \ b)] \ (a \ c)
where the latter expression must also equal (a ∧ c) ∧ b. Hence establishing B2 will also establish
B3. But
(a ∧ b) ∧ c =again [a \ (a \ c)] \ (a \ b) =B8 [a \ (a \ b)] \ [(a \ c) \ (a \ b)]
=B7 {a \ [(a \ c) \ (a \ b)]} \ (a \ b)
=B19 {a \ [a ∧ (b \ c)]} \ (a \ b) =B5 [a \ (a \ (a \ (b \ c)))] \ (a \ b)
=B15 [a \ (b \ c)] \ (a \ b) =B19 a ∧ [b \ ((b \ c))]
=B5 a ∧ (b ∧ c). £
Theorem 7.1.6 holds trivially for any iBCS-algebra arising as a reduct of an algebra in a
binary discriminator variety, for one begins with a generating class K of binary discriminator
algebras that implicitly satisfy B1-B5, where B5 defines ∧. From these algebras B1-B5 are
passed to all algebras in the variety. Although we do not show this here, the variety of iBCS-
algebras is in fact a binary discriminator variety (Bignall and Spinks [2007]). Theorem 7.1.6 is
thus a corollary to this fact. This theorem, however, takes us only halfway to characterizing
iBCS-algebras in terms of left normal bands with zero. Indeed:
Theorem 7.1.7. If (S; ∧, 0) is the derived left normal band with zero of an iBCS-algebra
(S; \, 0), then for each a ∈ S the set ⎡a⎤ = {b ∈S⎮b ≤ a} is both a subalgebra of S and a Boolean
lattice under the natural partial ordering of S.
Conversely, given a left normal band with zero (S; ∧, 0) such that for each a ∈S the set
⎡a⎤ is a Boolean lattice under ≥, a derived iBCS-structure (S; \, 0) is given by letting a \ b be the
relative complement of a∧b in ⎡a⎤ for all a, b in S.
Finally, both derivations (S; \, 0) → (S; ∧, 0) and (S; ∧, 0) → (S; \, 0) are reciprocal.
In particular, a \ b is the complement of a∧b in ⎡a⎤.
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=B13 (a \ [(a \ (b \ c)]) \ (c \ [a \ (b \ c)]) =B8 (a \ c) \ (a \ (b \ c))
=B7 (a \ [a \ (b \ c)]) \ c =B8 (a \ c) \ ([(a \ (b \ c)] \ c)
=B7 (a \ c) \ [(a \ c) \ (b \ c)] =B5 (a \ c) ∧ (b \ c) =B18 (a \ c) \ (a \ b). £
This leads us to:
Theorem 7.1.6. Given an iBCS-algebra (A; \, 0), upon setting x∧y = x \ (x \ y) the
derived algebra (A; ∧, 0) is a left normal band with zero.
Proof. B1 follows from a ∧ a = a \ (a \ a) = a \ 0 = a. B4 follows from
a ∧ 0 = a \ (a \ 0) = a \ a = 0 and 0 ∧ a = 0 \ (0 \ a) = 0
provided we know that 0 \ a = 0 holds in general. But the latter follows from 0 \ a = (a \ a) \ a =
(a \ a) \ (a \ a) = 0 \ 0 = 0. To verify B2 and B3 we again follow Spinks [2002]. To begin,
observe that
(a ∧ b) ∧ c =B5 (a ∧ b) \ [(a ∧ b) \ c] =B5, 16 [a \ (a \ b)] \ [(a \ c) \ (a \ b)]
=B8 [a \ (a \ c)] \ (a \ b) =B7 [a \ (a \ b)] \ (a \ c)
where the latter expression must also equal (a ∧ c) ∧ b. Hence establishing B2 will also establish
B3. But
(a ∧ b) ∧ c =again [a \ (a \ c)] \ (a \ b) =B8 [a \ (a \ b)] \ [(a \ c) \ (a \ b)]
=B7 {a \ [(a \ c) \ (a \ b)]} \ (a \ b)
=B19 {a \ [a ∧ (b \ c)]} \ (a \ b) =B5 [a \ (a \ (a \ (b \ c)))] \ (a \ b)
=B15 [a \ (b \ c)] \ (a \ b) =B19 a ∧ [b \ ((b \ c))]
=B5 a ∧ (b ∧ c). £
Theorem 7.1.6 holds trivially for any iBCS-algebra arising as a reduct of an algebra in a
binary discriminator variety, for one begins with a generating class K of binary discriminator
algebras that implicitly satisfy B1-B5, where B5 defines ∧. From these algebras B1-B5 are
passed to all algebras in the variety. Although we do not show this here, the variety of iBCS-
algebras is in fact a binary discriminator variety (Bignall and Spinks [2007]). Theorem 7.1.6 is
thus a corollary to this fact. This theorem, however, takes us only halfway to characterizing
iBCS-algebras in terms of left normal bands with zero. Indeed:
Theorem 7.1.7. If (S; ∧, 0) is the derived left normal band with zero of an iBCS-algebra
(S; \, 0), then for each a ∈ S the set ⎡a⎤ = {b ∈S⎮b ≤ a} is both a subalgebra of S and a Boolean
lattice under the natural partial ordering of S.
Conversely, given a left normal band with zero (S; ∧, 0) such that for each a ∈S the set
⎡a⎤ is a Boolean lattice under ≥, a derived iBCS-structure (S; \, 0) is given by letting a \ b be the
relative complement of a∧b in ⎡a⎤ for all a, b in S.
Finally, both derivations (S; \, 0) → (S; ∧, 0) and (S; ∧, 0) → (S; \, 0) are reciprocal.
In particular, a \ b is the complement of a∧b in ⎡a⎤.
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